Essential formulae
Number and Algebra
Laws of indices:
a
m
× a
n
= a
m+n a m
a n = a
m−n
(a
m
)
n
= a
mn
a
m
n =
n
√
a m
a −n =
1
a n
a 0 = 1
Quadratic formula:
If ax
2
+ bx + c = 0 then x =
−b ±
√
b 2 − 4ac
2a
Factor theorem:
If x = a is a root of the equation f (x) = 0, then (x − a)
is a factor of f (x).
Remainder theorem:
If (ax 2 + bx + c) is divided by (x − p), the
remainder will be: ap 2 + bp +c.
or if (ax 3 + bx 2 + cx + d) is divided by (x − p), the
remainder will be: ap 3 + bp 2 + cp + d.
Partial fractions:
Provided that the numerator f (x) is of less degree than
the relevant denominator, the following identities are
typical examples of the form of partial fractions used:
f (x)
(x + a)(x + b)(x + c)
≡
A
(x + a)
+
B
(x + b)
+
C
(x + c)
f (x)
(x + a) 3 (x + b)
≡
A
(x + a)
+
B
(x + a) 2 +
C
(x + a) 3 +
D
(x + b)
f (x)
(ax 2 + bx + c)(x + d)
≡
Ax + B
(ax 2 + bx + c)
+
C
(x + d)
Definition of a logarithm:
If y = a x then x = log a y
Laws of logarithms:
log(A × B) = log A + log B
log
A
B
= log A − log B
log A
n
= n × log A
Exponential series:
e
x
= 1 + x +
x 2
2!
+
x 3
3!
+ · · ·
(valid for all values of x)
Hyperbolic functions:
sinh x =
e
x
− e
−x
2
cosech x =
1
sinh x
=
2
e x − e −x
cosh x =
e
x
+ e
−x
2
sech x =
1
cosh x
=
2
e x + e −x
tanh x =
e x − e −x
e x + e −x
coth x =
1
tanh x
=
e x + e −x
e x − e −x
cosh
2 x − sinh
2
= 1 1 − tanh
2 x = sech
2 x
coth
2 x − 1 = cosech
2 x